已知復(fù)數(shù)z的輻角為60°,且|z-1|是|z|和|z-2|的等比中項.求|z|.
【答案】
分析:本題考查的復(fù)數(shù)的基本概念及等比數(shù)列的性質(zhì),由復(fù)數(shù)z的輻角為60°,我們可以使用待定系數(shù)法設(shè)出復(fù)數(shù)Z,然后根據(jù)|z-1|是|z|和|z-2|的等比中項,結(jié)合等比數(shù)列的性質(zhì)構(gòu)造方程,解方程求出待定的系數(shù),即可得到Z值,進而求出復(fù)數(shù)的模.
解答:解:設(shè)z=(rcos60°+rsin60°),
則復(fù)數(shù)z的實部為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213053415877110/SYS201310232130534158771010_DA/0.png)
.
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213053415877110/SYS201310232130534158771010_DA/1.png)
由題設(shè)|z-1|
2=|z|•|z-2|,
即:(z-1)(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213053415877110/SYS201310232130534158771010_DA/2.png)
-1)=|z|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213053415877110/SYS201310232130534158771010_DA/3.png)
∴r
2-r+1=r
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213053415877110/SYS201310232130534158771010_DA/4.png)
,
整理得r
2+2r-1=0.
解得r=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213053415877110/SYS201310232130534158771010_DA/5.png)
-1,
r=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213053415877110/SYS201310232130534158771010_DA/6.png)
-1(舍去).
即|z|=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213053415877110/SYS201310232130534158771010_DA/7.png)
-1.
點評:解決復(fù)數(shù)問題時,我們多使用待定系數(shù)法,即設(shè)出復(fù)數(shù)的值,然后根據(jù)題目中的其它條件,列出方程,解方程求出系數(shù),即可得到未知復(fù)數(shù)的值.